In this work, we further investigate the application of the well-known Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for numerically solving initial-value problems of systems of ordinary differential equations. By extending the ideas of our previous paper, we now utilize some advanced versions of RE in the form of repeated RE (RRE). Assume that the underlying LMM -- the base method -- has order $p$ and RE is applied $l$ times. Then we prove that the accelerated sequence has convergence order $p+l$. The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g. $A$- or $A(\alpha)$-stability) are preserved by the LMM-RGRE methods.

翻译：本文进一步研究了著名的Richardson外推（RE）技术在加速线性多步方法（LMMs）数值求解常微分方程组初值问题时序列收敛性中的应用。在前期工作的基础上，我们采用重复Richardson外推（RRE）的高级形式。假设底层LMM（基方法）具有p阶精度，且RE被应用l次，则证明加速后的序列具有p+l阶收敛性。本文提出的版本为全局RE（GRE，也称被动RE），因为线性组合项是独立计算的。因此，所得的高阶LMM-RGRE方法可并行实现，且现有LMM代码无需修改即可直接使用。此外，我们研究了LMM-RGRE方法如何保持基方法的线性稳定性性质（如A-稳定性或A(α)-稳定性）。